Victor Ginzburg

Koszul Duality Patterns in Representation Theory

The aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of Koszul duality to repre- sentation theory. The paper consists of three parts relatively independent of each other. The first part gives a reasonably selfcontained introduction to Koszul rings and Koszul duality. Koszul rings are certain Z-graded rings with particularly nice homological properties which involve a kind of duality. Thus, to a Koszul ring one associates naturally the dual Koszul ring. The second part is devoted to an application to representation theory of semisimple Lie algebras. We show License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use KOSZUL DUALITY PATTERNS 527 that the block of the Bernstein-Gelfand-Gelfand category O that corresponds to any fixed central character is governed by the Koszul ring. Moreover, the dual of that ring governs a certain subcategory of the category O again. This generalizes the selfduality theorem conjectured by Beilinson and Ginsburg in 1986 and proved by Soergel in 1990. In the third part we study certain cate- gories of mixed perverse sheaves on a variety stratified by affine linear spaces. We provide a general criterion for such a category to be governed by a Koszul ring. In the flag variety case this reduces to the setup of part two. In the more general case of affine flag manifolds and affine Grassmannians the criterion should yield interesting results about representations of quantum groups and affine Lie algebras. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 E-mail address: sasha@math.mit.edu Department of Mathematics, The University of Chicago, Chicago, Illinois 60637 E-mail address: ginzburg@math.uchicago.edu Max-Planck-Institut fur Mathematik, Gottfried-Claren-Strase 26, D-53 Bonn 3, Germany Current address: Mathematisches Institut, Universitat Freiburg, Albertstrase 23b, D-79104 Freiburg, Germany E-mail address: soergel@sun1.mathematik.uni-freiburg.de License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Koszul duality for operads

(0.1) The purpose of this paper is to relate two seemingly disparate developments. One is the theory of graph cohomology of Kontsevich [Kon 2 3] which arose out of earlier works of Penner [Pe] and Kontsevich [Kon 1] on the cell decomposition and intersection theory on the moduli spaces of curves. The other is the theory of Koszul duality for quadratic associative algebras which was introduced by Priddy [Pr] and has found many applications in homological algebra, algebraic geometry and representation theory (see e.g., [Be] [BGG] [BGS] [Ka 1] [Man]). The unifying concept here is that of an operad. This paper can be divided into two parts consisting of chapters 1, 3 and 2, 4, respectively. The purpose of the first part is to establish a relationship between operads, moduli spaces of stable curves and graph complexes. To each operad we associate a collection of sheaves on moduli spaces. We introduce, in a natural way, the cobar complex of an operad and show that it is nothing but a (special case of the) graph complex, and that both constructions can be interpreted as the Verdier duality functor on sheaves. In the second part we introduce a class of operads, called quadratic, and introduce a distinguished subclass of Koszul operads. The main reason for introducing Koszul operads (and in fact for writing this paper) is that most of the operads ”arising from nature” are Koszul, cf. (0.8) below. We define a natural duality on quadratic operads (which is

On the category O for rational Cherednik algebras

Abstract We study the category 𝒪 of representations of the rational Cherednik algebra AW attached to a complex reflection group W. We construct an exact functor, called Knizhnik-Zamolodchikov functor: 𝒪→ℋW-mod, where ℋW is the (finite) Iwahori-Hecke algebra associated to W. We prove that the Knizhnik-Zamolodchikov functor induces an equivalence between 𝒪/𝒪tor, the quotient of 𝒪 by the subcategory of AW-modules supported on the discriminant, and the category of finite-dimensional ℋW-modules. The standard AW-modules go, under this equivalence, to certain modules arising in Kazhdan-Lusztig theory of “cells”, provided W is a Weyl group and the Hecke algebra ℋW has equal parameters. We prove that the category 𝒪 is equivalent to the module category over a finite dimensional algebra, a generalized “q-Schur algebra” associated to W.

Quantization of Slodowy slices

We give a direct proof of (a slight generalization of) the recent result of A. Premet related to generalized Gelfand-Graev representations and of an equivalence due to Skryabin.

Finite-dimensional representations of rational Cherednik algebras

A complete classification and character formulas for finite-dimensional irreducible representations of the rational Cherednik algebra of type A is given. Less complete results for other types are obtained. Links to the geometry of affine flag manifolds and Hilbert schemes are discussed.

Almost-commuting variety, D-modules, and Cherednik algebras

We study a scheme M closely related to the set of pairs of n by n-matrices with rank 1 commutator. We show that M is a reduced complete intersection with n+1 irreducible components, which we describe. There is a distinguished Lagrangian subvariety Nil in M. We introduce a category, C, of D-modules whose characteristic variety is contained in Nil. Simple objects of that category are analogous to Lusztigs character sheaves. We construct a functor of Quantum Hamiltonian reduction from category C to the category O for type A rational Cherednik algebra.

Geometric methods in the representation theory of Hecke algebras and quantum groups

These lectures are mainly based on, and form a condensed survey of the book by N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry, Birkhauser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal enveloping algebra of a complex semisimple Lie algebra, a quantum group or the Iwahori-Hecke algebra of bi-invariant functions (under convolution) on a p-adic group, are considered. We give a uniform geometric construction of these algebras in terms of homology of an appropriate “Steinberg-type” variety Z (or its modification, such as K-theory or elliptic cohomology of Z, or an equivariant version thereof). We then explain how to obtain a complete classification of finite dimensional irreducible representations of the algebras in question, using our geometric construction and perverse sheaves methods.

Cherednik algebras and Hilbert schemes in characteristic

We prove a localization theorem for the type An−1 rational Cherednik algebra Hc = H1,c(An−1) over Fp, an algebraic closure of the finite field. In the most interesting special case where c ∈ Fp, we construct an Azumaya algebra Hc on Hilb n A2, the Hilbert scheme of n points in the plane, such that Γ(Hilb A2, Hc) = Hc. Our localization theorem provides an equivalence between the bounded derived categories of Hc-modules and sheaves of coherent Hc-modules on Hilb n A2, respectively. Furthermore, we show that the Azumaya algebra splits on the formal neighborhood of each fiber of the Hilbert-Chow morphism. This provides a link between our results and those of Bridgeland, King and Reid, and Haiman.

Principal nilpotent pairs in a semisimple Lie algebra 1

Abstract.This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The properties of these pairs and their role is similar to those of the principal nilpotents. Each principal nilpotent pair gives rise to a harmonic polynomial on the Cartesian square of the Cartan subalgebra, that transforms under an irreducible representation of the Weyl group. In the special case of ??n the conjugacy classes of principal nilpotent pairs and the irreducible representations of the symmetric group, Sn, are both parametrised (in a compatible way) by Young diagrams. In general, our theory provides a natural generalization to arbitrary Weyl groups of the classical construction of simple Sn-modules in terms of Young’s symmetrisers. First results towards a complete classification of all principal nilpotent pairs in a simple Lie algebra are presented at the end of this paper in an Appendix, written by A. Elashvili and D. Panyushev.

Wall-crossing functors and -modules

We study Translation functors and Wall-Crossing functors on infinite dimensional representations of a complex semisimple Lie algebra using D-modules. This functorial machinery is then used to prove the Endomorphism-theorem and the Structure-theorem, two important results established earlier by W. Soergel in a totally different way. Other applications to the category O of Bernstein-Gelfand-Gelfand are given, and some conjectural relationships between Koszul duality, Verdier duality and convolution functors are discussed. A geometric interpretation of tilting modules is given.